Math Research Experience: Surface Intersections

During the summer between my junior and senior year, I was extremely lucky to be one of the 80 rising seniors who were admitted to the Research Science Institute (RSI) held at MIT. My mentor was Ryan Reich, a graduate student from Harvard University. Before the program, I was contacted by my mentor and was given a couple of project ideas. I could either do a project on tropical geometry or do a project similar to that of his previous student. I chose the latter. His previous student studied the intersection of two quadric surfaces in CP and found an explicit normal form. This paper can be found at [9]. I thoroughly read through the paper and learned of things such as projective space, pencils, fractional linear transformations, and birational equivalence. After I finished reading this paper, I decided to read some of its references. I read a few chapters from Silverman and Tate’s Rational Points on Elliptic Curves ([6]) and Silverman’s Arithmetic of Elliptic Curves ([7]), and a couple of papers that were used by Q. Yuan in his paper such as [1], [4], [5]. Additionally I decided to learn some complex analysis from Brown and Churchill’s Complex Analysis and Applications ([2]). Through this I familiarized myself with elliptic curves and their group law, quadric surface intersection, the Weierstrass form of elliptic curves, and some more projective geometry. This whole process of learning more math took about a month before I left for RSI. A quadric surface and a cubic surface are defined at the beginning of Section 3. A normal form X for a certain set of objects S means that every element in S is equivalent to X. For example, (x− h) + (y − k) = r is a normal form for all circles.